Prevent mesh from going below the grid?
You could create a plane on Z = 0 (an empty will do, too, but you might want to hide what's below), then apply a Mirror modifier along Z around that plane, with Clipping enabled:

This will effectively clamp vertex positions:

If you leave a vertex at the plane, it will stick to it, and you will have to temporarily disable Clipping to be able to move it away: